Combinatorial representations Peter J. Cameron CAUL, Lisbon, April 2012 Joint work with Max Gadouleau and Søren Riis see arXiv 1109.1216 Think of the elements of a matroid as being a family (vi : i 2 E) of vectors in a vector space V. (It is a family rather than a set since we don’t mind if vectors are repeated.) A matroid can be described in many different ways: by the independent sets, the bases, the minimal dependent sets, the rank function . Matroids A matroid is a structure for describing the linear independence and dependence of sets of vectors in a vector space. A matroid can be described in many different ways: by the independent sets, the bases, the minimal dependent sets, the rank function . Matroids A matroid is a structure for describing the linear independence and dependence of sets of vectors in a vector space. Think of the elements of a matroid as being a family (vi : i 2 E) of vectors in a vector space V. (It is a family rather than a set since we don’t mind if vectors are repeated.) Matroids A matroid is a structure for describing the linear independence and dependence of sets of vectors in a vector space. Think of the elements of a matroid as being a family (vi : i 2 E) of vectors in a vector space V. (It is a family rather than a set since we don’t mind if vectors are repeated.) A matroid can be described in many different ways: by the independent sets, the bases, the minimal dependent sets, the rank function . I Algebraic matroids: E is a family of elements in a field with a prescribed algebraically closed subfield F, and independence means algebraic independence over F (so that rank is transcendence degree). I Graphic matroids: E is the edge set of a graph, and a set of edges is independent if it contains no circuit (so that the rank of a set A of edges is n − c(A), where n is the number of vertices and c(A) the number of connected components in the graph with edge set A. I Transversal matroids: E is the index set of a family of subsets of a set S, and a subset of E is independent if the corresponding subfamily possesses a transversal. Examples Vectors in a vector space form the standard examples of matroids. But the concept is important because there are many other examples: I Graphic matroids: E is the edge set of a graph, and a set of edges is independent if it contains no circuit (so that the rank of a set A of edges is n − c(A), where n is the number of vertices and c(A) the number of connected components in the graph with edge set A. I Transversal matroids: E is the index set of a family of subsets of a set S, and a subset of E is independent if the corresponding subfamily possesses a transversal. Examples Vectors in a vector space form the standard examples of matroids. But the concept is important because there are many other examples: I Algebraic matroids: E is a family of elements in a field with a prescribed algebraically closed subfield F, and independence means algebraic independence over F (so that rank is transcendence degree). I Transversal matroids: E is the index set of a family of subsets of a set S, and a subset of E is independent if the corresponding subfamily possesses a transversal. Examples Vectors in a vector space form the standard examples of matroids. But the concept is important because there are many other examples: I Algebraic matroids: E is a family of elements in a field with a prescribed algebraically closed subfield F, and independence means algebraic independence over F (so that rank is transcendence degree). I Graphic matroids: E is the edge set of a graph, and a set of edges is independent if it contains no circuit (so that the rank of a set A of edges is n − c(A), where n is the number of vertices and c(A) the number of connected components in the graph with edge set A. Examples Vectors in a vector space form the standard examples of matroids. But the concept is important because there are many other examples: I Algebraic matroids: E is a family of elements in a field with a prescribed algebraically closed subfield F, and independence means algebraic independence over F (so that rank is transcendence degree). I Graphic matroids: E is the edge set of a graph, and a set of edges is independent if it contains no circuit (so that the rank of a set A of edges is n − c(A), where n is the number of vertices and c(A) the number of connected components in the graph with edge set A. I Transversal matroids: E is the index set of a family of subsets of a set S, and a subset of E is independent if the corresponding subfamily possesses a transversal. Matroids can be axiomatised in this way. The crucial axiom is the exchange axiom: If B and C are bases and e 2 B n C, then there exists f 2 C n B such that B n feg [ ff g is a basis. It follows that any two bases have the same size, the rank of the matroid. I will use bases to describe matroids and similar structures in this talk. Matroid bases A matroid is completely specified by its bases, that is, its maximal independent sets. If B and C are bases and e 2 B n C, then there exists f 2 C n B such that B n feg [ ff g is a basis. It follows that any two bases have the same size, the rank of the matroid. I will use bases to describe matroids and similar structures in this talk. Matroid bases A matroid is completely specified by its bases, that is, its maximal independent sets. Matroids can be axiomatised in this way. The crucial axiom is the exchange axiom: It follows that any two bases have the same size, the rank of the matroid. I will use bases to describe matroids and similar structures in this talk. Matroid bases A matroid is completely specified by its bases, that is, its maximal independent sets. Matroids can be axiomatised in this way. The crucial axiom is the exchange axiom: If B and C are bases and e 2 B n C, then there exists f 2 C n B such that B n feg [ ff g is a basis. I will use bases to describe matroids and similar structures in this talk. Matroid bases A matroid is completely specified by its bases, that is, its maximal independent sets. Matroids can be axiomatised in this way. The crucial axiom is the exchange axiom: If B and C are bases and e 2 B n C, then there exists f 2 C n B such that B n feg [ ff g is a basis. It follows that any two bases have the same size, the rank of the matroid. Matroid bases A matroid is completely specified by its bases, that is, its maximal independent sets. Matroids can be axiomatised in this way. The crucial axiom is the exchange axiom: If B and C are bases and e 2 B n C, then there exists f 2 C n B such that B n feg [ ff g is a basis. It follows that any two bases have the same size, the rank of the matroid. I will use bases to describe matroids and similar structures in this talk. Let E be the ground set and B the family of bases of a matroid M of rank r.A vector representation of M is an r assignment of a vector vi 2 F to each i 2 E, such that, for i1,..., ir 2 E, r (vi1 ,..., vir ) is a basis for F , fi1,..., irg 2 B. Matroid representations Now a matroid representation is simply described: Matroid representations Now a matroid representation is simply described: Let E be the ground set and B the family of bases of a matroid M of rank r.A vector representation of M is an r assignment of a vector vi 2 F to each i 2 E, such that, for i1,..., ir 2 E, r (vi1 ,..., vir ) is a basis for F , fi1,..., irg 2 B. r Notation: if fi1 ,..., fir : F ! F, then we regard the r-tuple r r (fi1 ,..., fir ) as being a function from F to F . Now a vector representation of the matroid M is an assignment r of a linear map fi : F ! F to each i 2 E, so that r r (fi1 ,..., fir ) : F ! F is a bijection , fi1,..., irg 2 B. in dual form Now regard the representing vectors v1,..., vr as lying in the r dual space of F . To emphasise this I will write fi instead of vi; r thus fi is a function from F to F. Now a vector representation of the matroid M is an assignment r of a linear map fi : F ! F to each i 2 E, so that r r (fi1 ,..., fir ) : F ! F is a bijection , fi1,..., irg 2 B. in dual form Now regard the representing vectors v1,..., vr as lying in the r dual space of F . To emphasise this I will write fi instead of vi; r thus fi is a function from F to F.